The assignee's prior document FR-A-2 718 347=EP-A-0 676 158 discloses a method and apparatus for supporting an element to be supported, in particular the body of a patient, making it possible to support the element at an essentially constant controlled penetration depth by means of a measurement device comprising an induction system combined with a piece of metal foil situated under the body being supported, the metal foil being displaced by penetration of the body being supported. The support device may comprise a single chamber, as shown in FIG. 1 of that document, and reproduced in accompanying FIG. 1. Alternatively, as shown in FIG. 3 of the assignee's prior document, provision is made for a support device that has three chambers 114a, 114b, 114c including a chamber for the heel region, each chamber being provided with a respective measurement device for measuring penetration depth, so as to enable each chamber to be individually controlled to remain at a suitable respective inflation pressure. That procedure complicates the apparatus and its method of operation.